Calculation of 6j symbols
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
At present there is no complete algorithm for the calculation of the coupling or recoupling factors of an arbitrary compact group. However, this thesis is based on the premise that the 6j can be calculated using only the Kronecker product rules for the group and the general relations between 6j. We review the symmetry properties of the 6j symbols and choose a set of values for the permutation matrices of a mixed symmetry triad. An algorithm is presented for the recursive calculation of coupling and recoupling factors in terms of the primitive factors. This algorithm is shown to be complete. It is then shown how this algorithm may be applied to a larger class of group theoretic transformation factors. The primitive 6j are then split into four classes. This allows us to specify complete algorithms for the calculation of all but one of these classes. This is a major advance since it was previously necessary to systematically try all equations in order to solve an unknown 6j. We conjecture that our algorithm for the calculation of the fourth class of primitive 6j, the core 6j, is complete, although we are only able to prove this for SO₃. As a consequence we discuss the various special cases that occur in groups more complex than SO₃, starting with the point groups. New results are given for the groups G₂ and E₈, and the 6j for the mixed symmetry finite group K₂₀ are completely solved. The data structures necessary for the implementation of the algorithms in a PASCAL program are discussed, along with the algorithms required to calculate the symmetrised powers of an irrep.